Does rectifiable curve in $\mathbb{R^n}$ must be piecewise smooth?

Question

For a curve in $n$ dimension Euclid Space $\mathbb{R}^n$, if it is rectifiable, then does it have to be a smooth curve, or piecewise smooth.

And of course, how to prove it.

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To define smooth, there exists $\vec{x}(t)=(x_1(t),...,x_n(t))^T$ and $\frac{dx_i(t)}{dt}$ exists and continues. Piecewise smooth saying that by saperate curve into limite parts, each part will have the above condition.

To define length, saying we use points $P=\{\vec{x}_1,...,\vec{x}_{m+1}\}$ saperate the curve into $m$ parts, donate:

1. The distance $\Delta \vec{x}_i=\sqrt{\sum_{j=1}^n(x_{(j)i}-x_{(j)i+1}}$
2. $\lambda=\max(\Delta x_i)$

Then if dispite of ways of partition, that is $\forall P$ we always have $\lim_{\lambda\to0}\sum_{i=1}^m\Delta\vec{x}_i=L<\infty$ then it have length and the length is $L$

Answer 1

A curve $t\mapsto\big(x_1(t),\ldots,x_n(t)\big)$ is rectifiable if and only if $x_1(t),\ldots,x_n(t)$ are of bounded variation. So, the set of points where it is not smooth may well be infinite (and even uncountable; consider the graph of the Cantor function).